The Hill Coefficient

Thus the Hill coefficient is essentially a measure of the sensitivity of the binding saturation to the substrate concentration. For the saturation curve of Equation (4.33), h = 2, corresponding to theoretical maximum for the case of two binding sites. 

The previous section illustrated how allosteric cooperativity can result in a sigmoidal relationship between binding saturation and substrate concentration. In this section, we demonstrate how a sigmoidal relationship between product concentration and time can arise from enzyme kinetics with time lags. 

Recall that in the standard Michaelis-Menten enzyme kinetics we approximate the kinetics of substrate and product using Equation (3.32) or (4.26) for the essentially irreversible case  

Thus the initial phase of the [P] as a function of t is an increasing function with negative curvature. However, for certain enzyme kinetics, [P] as a function of t has a positive curvature, a lag, in its initial phase. This phenomenon is known as hysteresis, first discovered by Carl Frieden [60], 

Strictly speaking, any multi-step kinetic scheme will involve a lag. However, realistically observing hysteresis in enzyme kinetics is always associated with the existence of one of several slow step(s) prior to the final step. This is because if all the steps prior to the final step were fast, then there would be a rapid pre-equilibriation and the rapid steps could be lumped into a single kinetic species (see Section 4.2.1). 

To account for differences in the Hill coefficient, enzyme inhibition data are best ht to Equation (5.4) or (5.5). In measuring the concentration-response function for small molecule inhibitors of most target enzymes, one will hnd that the majority of compounds display Hill coefficient close to unity. However, it is not uncommon to hnd examples of individual compounds for which the Hill coefficient is signihcandy greater than or less than unity. When this occurs, the cause of the deviation from expected behavior is often reflective of non-ideal behavior of the compound, rather than a true reflection of some fundamental mechanism of enzyme-inhibitor interactions. Some common causes for such behavior are presented below. 

The second common cause of a low Hill coefficient is a partitioning of the inhibitor into an inactive, less potent, or inaccessible form at higher concentrations. This can result from compound aggregation or insolubility. As the concentration of compound increases, the equilibrium between the accessible and inaccessible forms may increase, leading to a less than expected % inhibition at the higher concentrations. This will tend to skew the concentration-response data, resulting in a poorer 

The concentration of monomer present at any concentration of inhibitor is given by SC, and the concentration of dimer is given, considering mass balance, by (1 - 8)C. When an enzyme is treated simultaneously with two inhibitors, / and J, that bind in a mutually exclusive fashion, the fractional activity is given by (Copeland, 2000) 

If I represents the monomer and J represents the dimer of our inhibitory molecule, then Equation (5.7) becomes 

In this case, fitting the concentration-response data to Equation (5.4) would yield a smooth curve that appears to ht well but with a Hill coefficient much less than unity. 

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The most general equation for the correlation between effect (E) and concentration (C) is given by the sigmoid inax model where the concentration (CE50) produces the half-maximum effect and the Hill coefficient (H) specifies the sigmoidicity (Fig. 2). 

If the Hill coefficient is less than one (H < 1), the maximum effect will never be obtained with increasing concentrations (E < max). If the Hill coefficient is more than ten (H > 10), an on-off phenomenon can be described at CE50. The effect can be assumed to be marginal and less than 5% of max for the threshold concentration (CE05) but almost 95% of max for the ceiling concentration (CE95). 

A linear form of the Hill equation is used to evaluate the cooperative substrate-binding kinetics exhibited by some multimeric enzymes. The slope n, the Hill coefficient, reflects the number, nature, and strength of the interactions of the substrate-binding sites. A... 

DERIVATION OF THE HILL COEFFICIENT (OR SLOPE) AS A DETERMINANT OF THE NUMBER OF BINDING SITES FOR AN AGONIST (NEUROTRANSMITTER) ON ITS RECEPTOR... 

In all these situations the Hill coefficient provides a warning sign to the medicinal chemist that the physical properties of the compound may render it intractable for further consideration. In short, whenever the Hill coefficient is significantly different from unity, the experimental data and the quality of the lead compound must be scrutinized much more carefully. 

For example, if the Hill coefficient (h) is unity, and we wish to achieve 25% inhibition, the fraction velocity would be 0.75, and its reciprocal (voM) would be 1.33. Plugging this into Equation (5.9), we find that 25% inhibition is obtained at a concentration of inhibitor equal to 1/3 IC50. Table 5.3 summarizes the four inhibitor concentrations needed to achieve the desired inhibition levels (again, at [5] = KM) when the Hill coefficient is unity and 3.0. 

Hence, a plot of log (pAR /(I -pAR)) against log [A] should give a straight line with a slope of one. Such a graph is described as a Hill plot, again after A. V. Hill, who was the first to employ it, and it is often used whenpAR is measured directly with a radiolabeled ligand (see Chapter 5). In practice, the slope of the line is not always unity, or even constant, as will be discussed. It is referred to as the Hill coefficient (%) the term Hill slope is also used. 

This is usually described as the Hill equation (see also Appendix 1.2C [Section 1.2.4.3]). Here, H is again the Hill coefficient, and y and vm l, are, respectively, the observed response and the maximum response to a large concentration of the agonist, A. [A]50 is the concentration of A at which y is half maximal. Because it is a constant for a given concentration-response relationship, it is sometimes denoted by K. While this is algebraically neater (and was the symbol used by Hill), it should be remembered that K in this context does not necessarily correspond to an equilibrium constant. Employing [A]50 rather than K in Eq. (1.6) helps to remind us that the relationship between... 

Hill plots are often used in pharmacology, where y may be either the fractional response of a tissue or the amount of a ligand bound to its binding site, expressed as a fraction of the maximum binding, and x is the concentration. It is sometimes found (especially when tissue responses are measured) that the Hill coefficient differs markedly from unity. What might this mean ... 

Pbmd = KHl)KA 2) + 2tfA(2)[A] + (1 + E)[Af The Hill plot would again be nonlinear with the Hill coefficient given by ... 

The Hill plot is log (B (Bnu>. - B)) vs. log [L], As noted earlier, the slope of the Hill plot (the Hill coefficient, H) is of particular utility. If the equation holds, a straight line of slope = 1 should be obtained. A value greater than 1 may indicate positive cooperativity, and a slope less than 1 either negative cooperativity or commonly the presence of sites with different affinities. The data of Problem 5.1 are also presented as a Hill plot in Figure 5.10. 

In Chapter 1 (Section 1.2.4.3), the Hill equation and the Hill coefficient, nH, are described. Hill coefficients greater than or less than unity are often interpreted as indicating positive or negative cooperativity, respectively, in the relationship between receptor occupancy and response. For example, positive cooperativity could arise due to amplification in a transduction mechanism mediated by G-proteins and changes in cell calcium concentration. 

Similar to Eq. (67), the first reaction (incorporating the enzyme phosphofructo-kinase) exhibits a Hill-type inhibition by its substrate ATP [126]. The overall ATP utilization v3 (ATP) is modeled by a saturable Michaelis Menten function. The system is specified by five kinetic parameters (with Gx lumped into Vm ), the Hill coefficient n, and the total concentration, 4 / = [ATP] + [ADP]. Note that the model is not intended to capture biological realism, rather it serves as a paradigmatic example to identify dynamic behavior in metabolic pathways. 

For the irreversible reactions, we assume Michaelis Menten kinetics, giving rise to 15 saturation parameters O1. C [0, 1] for substrates and products, respectively. In addition, the triosephospate translocator is modeled with four saturation parameters, corresponding to the model of Petterson and Ryde-Petterson [113]. Furthermore, allosteric regulation gives rise to 10 additional parameters 7 parameters 9" e [0, — n for inhibitory interactions and 3 parameters 0" [0, n] for the activation of starch synthesis by the metabolites PGA, F6P, and FBP. We assume n = 4 as an upper bound for the Hill coefficient. 

In graded response one can resort to using surrogates, and the classic Hill model (or sigmoid model as described in Equation 18.16 above) is used to correlate observed effect (E(t)) with the concentration modified by an exponent that is called Hill coefficient in classic sigmoidal model, the Hill coefficient h) would be equal to 1 ... 

The shape of curve is modulated by varying the Hill coefficient to fit the observed behavior. Sensitivity of a drug is how concentration translates into effect as described by the shape coefficient or the Hill coefficient at a value of 1, it is a classical parabola. Eg is... 

Comparison of Equafion 18.33 wifh Equation 18.29 shows that a is ln(EC5o) x is In C, and p corresponds to the Hill coefficient, h, which now represents interpatient variability. 

The reaction of PHGPx toward each substrate was analyzed by a HiU plot (Figure 7). The Hill coefficient was calculated to be 2.33 for cardiolipin monohydroperoxide and 1.37 for dihnoleoyl phosphatidylcholine... 

Another graphical method is the so-called Hill plot, which is a plot of log[0/( 1 - 0)] as a function of log x. The Hill coefficient is defined by... 

Clearly, the quantity = Xj/j) maps the region 0 < S < < into the interval 0 < S 2. The value of % = 2 is the maximum value of the Hill coefficient for the case m-l. One should be careful, however, to note that these particular methods are valid only for the case of two sites. When m > 2 there are various types of cooperativities and, in general, there is no single parameter that describes the cooperativity in the system. Even for the case m = 2 one could be misled in estimating the cooperativity of the system if one were to rely only on the/orm or the shape of the BI or any of its transformed functions, as will be demonstrated in Section 4.6 and again in Section 4.8 and Appendix F. 

Finally, we note that the fact that we have four different correlations in this system, some possibly of different signs, renders meaningless the characterization of the cooperativity of the system by a single number (as is frequently done using the Hill coefficient, see Section 4.3). We shall introduce in Section 5.8 a measure of the average cooperativity in a system, a quantity that may vary widely, even in its sign, as the binding process proceeds. 

Where equals the number of sites determined by some other biophysical procedure, we say that the system shows infinite cooperativity. No such behavior has been rigorously demonstrated for an enzyme or receptor. In the case of hemoglobin oxygenation (Fig. 2) under physiologic conditions, the Hill coefficient has a value of about 2.8. Of course, from X-ray structural information, we know that hemoglobin has four sites. Thus, we... 

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